Transcript EXAMPLE 1 Standardized Test Practice SOLUTION

EXAMPLE 1
Standardized Test Practice
SOLUTION
Let ( x1, y1 ) = ( –3, 5) and ( x2, y2 ) = ( 4, – 1 ).
d =
=
( x2 – x1 ) 2 + ( y2 – y1 ) 2
=
(4 – (–3))2 + (– 1 – 5)2
49 + 36
=
85
ANSWER
The correct answer is C.
EXAMPLE 2
Classify a triangle using the distance formula
Classify ∆ABC as scalene, isosceles, or equilateral.
AB = (7 – 4)2 + (3 – 6)2 = 18 = 3
BC = (2 – 7)2 + (1 – 3)2 =
29
AC = (2 – 4)2 + (1 – 6)2 =
29
2
ANSWER
Because BC = AC, ∆ABC is isosceles.
EXAMPLE 3
Find the midpoint of a line segment
Find the midpoint of the line segment joining (–5, 1)
and (–1, 6).
SOLUTION
Let ( x1, y1 ) = (–5, 1) and ( x2, y2 ) = (– 1, 6 ).
(
x1 + x2 , y1 + y2
2
2
)=(
– 5 + (– 1)
2
,
1+6
2
)=(
7
– 3,
2
)
EXAMPLE 4
Find a perpendicular bisector
Write an equation for the perpendicular bisector of the
line segment joining A(– 3, 4) and B(5, 6).
SOLUTION
STEP 1
Find the midpoint of the line
segment.
(
x1 + x2 , y1 + y2
2
2
)=(
–3 + 5
2
,
4 + 6
2
) = (1, 5)
EXAMPLE 4
Find a perpendicular bisector
STEP 2
Calculate the slope of AB
y2 – y1
6 – 4
m = x –x =
= 2 = 1
5 – (– 3)
4
8
2
1
STEP 3
Find the slope of the perpendicular bisector:
– 1 = – 1 =–4
m
1/4
EXAMPLE 4
Find a perpendicular bisector
STEP 4
Use point-slope form:
y – 5 = – 4(x – 1), or y = – 4x + 9.
ANSWER
An equation for the perpendicular bisector of AB is
y = – 4x + 9.
EXAMPLE 5
Solve a multi-step problem
Asteroid Crater
Many scientists believe that an asteroid slammed into
Earth about 65 million years ago on what is now
Mexico’s Yucatan peninsula, creating an enormous
crater that is now deeply buried by sediment. Use the
labeled points on the outline of the circular crater to
estimate its diameter. (Each unit in the coordinate
plane represents 1 mile.)
EXAMPLE 5
Solve a multi-step problem
SOLUTION
STEP 1
Write equations for the perpendicular bisectors of AO
and OB using the method of Example 4.
y = – x + 34
Perpendicular bisector of AO
y = 3x + 110
Perpendicular bisector of OB
EXAMPLE 5
Solve a multi-step problem
STEP 2
Find the coordinates of the center of the circle, where
AO and OB intersect, by solving the system formed by
the two equations in Step 1.
y = – x + 34
3x + 110 = – x + 34
Write first equation.
Substitute for y.
Simplify.
4x = – 76
x = – 19
Solve for x.
y = – (– 19) + 34 Substitute the x-value into the first
equation.
Solve for y.
y = 53
The center of the circle is C (– 19, 53).
EXAMPLE 5
Solve a multi-step problem
STEP 3
Calculate the radius of the circle using the distance
formula. The radius is the distance between C and any
of the three given points.
OC =
(–19 – 0)2 + (53 – 0)2 =
3170
56.3
Use (x1, y1) = (0, 0) and (x2, y2) = (–19, 53).
ANSWER
The crater has a diameter of about 2(56.3) = 112.6 miles.
EXAMPLE 1
Graph an equation of a parabola
Graph x = – 1 y2. Identify the focus, directrix, and axis
8
of symmetry.
SOLUTION
STEP 1
Rewrite the equation in standard form.
x = –1
8
– 8x = y2
Write original equation.
Multiply each side by – 8.
EXAMPLE 1
Graph an equation of a parabola
STEP 2
Identify the focus, directrix, and axis of symmetry. The
equation has the form y2 = 4px where p = – 2. The focus
is (p, 0), or (– 2, 0). The directrix is x = – p, or x = 2.
Because y is squared, the axis of symmetry is the
x - axis.
STEP 3
Draw the parabola by making a table of values and
plotting points. Because p < 0, the parabola opens to
the left. So, use only negative x - values.
EXAMPLE 1
Graph an equation of a parabola
EXAMPLE 2
Write an equation of a parabola
Write an equation of the parabola shown.
SOLUTION
The graph shows that the vertex is (0, 0) and the
3
directrix is y = – p = – 2 for p in the standard form of
the equation of a parabola.
x2 = 4py
Standard form, vertical axis of symmetry
( )
3
x2 = 4 2 y
x2 = 6y
Substitute
Simplify.
3
for p
2
EXAMPLE 3
Solve a multi-step problem
Solar Energy
The EuroDish, developed to provide electricity in
remote areas, uses a parabolic reflector to
concentrate sunlight onto a high-efficiency engine
located at the reflector’s focus. The sunlight heats
helium to 650°C to power the engine.
•
Write an equation for the
EuroDish’s cross section
with its vertex at (0, 0).
•
How deep is the dish?
EXAMPLE 3
Solve a multi-step problem
SOLUTION
STEP 1
Write an equation for the cross section. The engine is
at the focus, which is | p | = 4.5 meters from the vertex.
Because the focus is above the vertex, p is positive,
so p = 4.5. An equation for the cross section of the
EuroDish with its vertex at the origin is as follows:
x2 = 4py
Standard form, vertical axis of symmetry
x2 = 4(4.5)y
Substitute 4.5 for p.
x2 = 18y
Simplify.
EXAMPLE 3
Solve a multi-step problem
STEP 2
Find the depth of the EuroDish. The depth is the
y - value at the dish’s outside edge. The dish extends
8.5 = 4.25 meters to either side of the vertex (0, 0),
2
so substitute 4.25 for x in the equation from Step 1.
vertex (0, 0), so substitute 4.25 for x in the equation
from Step 1.
x2 = 18y
Equation for the cross section
(4.25)2 = 18y
Substitute 4.25 for p.
1.0
y
Solve for y.
EXAMPLE 3
Solve a multi-step problem
ANSWER
The dish is about 1 meter deep.